Electromechanical wave filter



Feb. 11, 1941- R. B. 'B'LACKMAN ETAL ELECTROMECHANICAL WAVE FILTER Filed March a, 1959 FIG. 64

9 Sheets-Sheet 1 2 FIG. 8A FIG. 83

K2 55 56 63 64 F/Gflm M lffli lF- h C a A, k, 5?

R B BLACKMAN 2 4 INVENTORS By 5. mmros 'T'TORNEV 11, 1941- I R. B. BLACKMAN ETAL 3 ELECTROMECHANI CAL WAVE FILTER Filed March 8,1959 9 Sheets-Sheet 6 FIG 28B F/G. 308 m MAI-Q3 g} l RBBLAC/(MA/V S5 J fELA/(ATOS A B zv-M ATTORNEP F v F "F 1941- R. B. BLACKMAN ETAL 2,231,404

' ELECTROMECHANICAL WAVE FILTER Filed larch 8, 1939 9 Sheets-Sheet 7 FIG, 36'

DISS/PA 71v; LINE Fla. .3414 wa 2:52::

'F I s 1 FIG. 34B q RB. BLACK/VAN a? j WVENTORSE LAKATOS o/s sma THZE 5y A TTORNEK MATERIAV v Feb. 1941- R. a. BLACKMAN ErAL 1,404

ELECTROMECHANICAL WAVE FILTER I Filed larch 8, 1939 9 Sheets-Sheet a FIG. 38A H6394 HMAP RECEIVING COILS ELECTRO-DYNAM/C I20 FIG. 40 V g 90 I i I O i 6, 16 f2 15, 1g w FREQUENCY 260 260 3022 304 274 27a F1534 We? 12 2 3/2 3043o2 262 T262 264 306 306' 264 v 0 c a c o o a 12M 1 I W 3 FIG. 4/8

264T 27 T264 26 v 04 .B.BLACKMAN INVENTORSE LAKATOS ATTORN Y Patented Feb. 11, 1941 ELECTROMECHANICAL WAVE FILTER Ralph B. Blackman, Cranford, N. .L, and Emory Lakatos, New York, N. Y., assignors to Bell Telephone Laboratories,

Incorporated, New

York, N. Y., a corporation of New York Application March 8, 1939, Serial No; 260,544

27 Claims. (01. 178-44) This invention relates to mechanical vibratory systems and novel methods of designing, constructing and utilizing the same. More particularly, it relates to such systems employing three-dimensional rigid masses and to novel methods of designing and constructing electromechanical and mechanical wave filters, networks and systems having transmission, phase and impedance characteristics comparable to those of a large number of conventional types of electrical wave filters and networks.

It has been realized for some time that for the subaudible and the lower audible frequencies, lumped-element type mechanical wave filters and vibratory systems offer great advantages over corresponding electrical devices especially where narrow frequency bands are involved. There have been indications also that such structures could be used to advantage throughout the lower portion of the carrier-frequency range, extending somewhat into the superaudible frequency range. V

There are two principal factors which create a potential superiority of mechanical over elec-' trical frequency selective systems in thelow frequency range. The first is the purely fortuitous circumstance that for the former the necessary element values can be supplied by convenient physical sizes of masses and elastic members, while for the latter the inductances and capacitances are inconveniently large.

With regard to carrier frequency systems, while the mechanical elements tend to require inconveniently small dimensions, nevertheless this disadvantage, as compared with electrical systems, is generally more than offset by the marked economy in space that obtains through the use of mechanical systems. The second factor,;which holds with greater force for the low frequency range, is that by proper design mechanical elements can be made with only a fractional part of the damping or energy dissipation that exists, even in well-designed coils. These factors indicate that it should be possible to build mechanical systems, such asfilters which are extremely compact in comparison with equivalent electrical ones and which have sharper cutoffs and greater attenuation in the neighborhood of the attenuation peak frequencies.

The use of mechanical vibratory systems for wave filters and the like devices has not hitherto been as widespread as the above possibilities would appear to indicate that it might be, principally for the reason that heretofore no practicable structures have been proposed for ob- ,5 taining characteristics as flexible as those obtainable by the compositing of the several wellknown series of electrical structures. As a practical matter, the types of mechanical construction which have been used are with few, if any, 10 exceptions equivalent to a sequence of simple L- type electrical sections. While it is true that more general structures have been proposed, these latter have not in general been mechanically practicable. They usually require ideal; hinges, massless levers, point masses and similar elements for the attainment of which no practical methods have heretofore been proposed.

By way of illustration of the principles of this invention, it will be shown that it is possible to obtain practical mechanical vibratory systems which possess dynamical properties corresponding to the well-known series of ladder-type electrical wave filter sections. Furthermore, exam-: ples will be given of how the mechanical structures can be combined so that their over-all characteristics shall be the equivalent of composited electrical wave filter structures.

In the mechanical vibratory systems to be de- 30 scribed, an extensive use is made of the properties of rigid bars. Therefore, before proceeding with the description of the specific structures, the theory of rigid bars as applied in this invention will first be described and the specific structures will be then immediately derivable.

Throughout this application, including the appended claims, the term rigid when applied to a bar or mass is to be interpreted as substantially invariable in shape and size under the application of all forces to which it will be subjected in the course of normal operation of the structure in which the bar or mass is incorporated. This interpretation is further to be taken to preclude the use of a bar or mass of such dimensions and resilience that natural resonance of the bar or mass per se should occur within or near the range of frequencies of the vibratory energy to be employed in connection with the structures of the invention, since shape and size vary under the tremendous stresses developed by resonant phenomena. Fortunately, the masses and mass distributions desirable for rigid bars employed in the structures of the 5 invention are such that natural resonant phenomena of the barsper se occur only at frequencies far removed from the frequency range of interest, for all materials such as iron, steel, brass, etc. which are more conveniently employed in constructing such bars. The structures of this invention therefore difier substantially in character from. numerous mechanical vibratory systems of the prior art in which the natural resonant phenomena of the bars or masses per se designedly contribute to the operating characteristics of the system. Accordingly, through out the following description, whenever bars or masses are mentioned it is to be understood... that they are rigid within 'thpurviewof the above description of the term; The theory of bars is necessarily based upon certain fundamental principles of mechanics. A

each is the well-known center of percussion with respect to the other; When a bar is supported at two such points a lateral force applied at one of the points will produce only pure rotation about the second of the points. Thebody may therefore oscillate about the second point without transmitting energy to the support at that point provided only that the support will permit free rotation within the rangerequired for the oscillatory motion. For this'condition, in parlance analogous to that employed in dealing with electrical circuits, there is no mutual mass coupling the points together and the inertia or inductive reactance of the bar to lateral oscillatory motion, within the limits above desig- (Dnated, may be freely utilized to provide, in effect, virtually independent point masses at either or both points of support. I

Conversely for any two points on the bar which are not mutually centers of percussion? with 45 respect to each other there will exist'a mutual mass coupling the points together. and the ap plication of lateral oscillatory energy at' one point will result in the transmission of like energy to the other, the transmission being dependent 50 upon the coupling between the points which in turn is determined by the mass distribution throughout thebar and the location of the, two points thereon.

In electrical circuit parlance, the bar is then 55 a transformer having particular finpu and output reactances (inertias) coupled by a mutual reactance (or inertia) Since the second point of support, of any pair'sele'cted, maybe chosen so as to be between the point which does 60 bear the relation of center of percussion with respect to the first point, and the firstpoint, or the second point may be chosen so as tobe beyond the center of percussion, so defined, it is obvious that the mutual reactance" coupling 65 the two points may be made such that the motion transmitted to the second point will be in phase, or 180 degrees out of phase, with the motion at the first point.

A permissible electrical analogue of the general bar supported at two points is therefore apparently a T-network of inductances, the series arms representing the effective inertias at the two points of support, respectively, .and the shunt arm representingthe .mutualf inertia coupling the two points.

In the detailed analysis given hereunder, formulae are developed by which the properties of any particular bar may be computed, numerous combinations of bars and resilient members are suggested and are shown to be capable of providing mechanical vibratory systems having properties equivalent to a large number of wellknown types of electrical networks and wave filter sections.

Methods of compositing mechanical sections of this invention are also explained and analyzed and it will be apparent from the analysis given that-the principles of the present invention permit the construction of mechanical wave filter sections and composited multisection mechanical "wave filters having properties equivalent to virtually any 'knownvariety of electrical laddervtype wave filter.

Obviously, also, the principles of the invention mayQbeIappIied to obtain the mechanical equivalent of practically any complex electrical network which involves reactance.

particularly simple relation exists when a bar or mass is supported at two points so related that 'I'he'extension of the field of practical utility for mechanical vibratory systems afforded by the principles of the present invention will be pointed out inmore detail hereinunder. description of the general bar theory will be followed by a description of the mechanical equivalents of a well-known series of ladder-type electrical wave filter sections including a number requiring negative mutual inductance. The illustrations are of a diagrammatic nature and should not be taken as indicating that the particular. type of construction suggested in any particular instance is the sole practicable one to be found. It will be shown that the majority of the well-known sections can readily be constructed as entities and further that mechanical structures of this invention may readily be made the equivalent of a composited multisection electrical structure such as a multisection electrical wave filter. Since, as will be hereinafter demonstrated,a variety of combinations of mechanical filter sections can arise from slight changes in arrangement, the technique disclosed hereinafter must be followed precisely.

For filters in which high attenuations of unwanted frequency ranges are desired, the elimination of parasitic transmission through the frame becomes of paramount importance. This problem will also be discussed and a number of applications of the general bar theory will be suggested as aids in its solution.

An object of this invention, therefore, is to facilitate the practical design of mechanical vibratory systems by the application of the general bar theory.

A further object is to provide practicable mechanical wave filter sections having properties closely analogous to those of numerous wellknown electrical wave filter sections. A further object is to provide a series of mechanioal wave filter sections having different attenuation characteristics but similar terminal impedance characteristics so that two or more sections may be readily composited to provide complex attenuation properties without introducing deleterious energy reflection or interaction effects between adjacent sections.

A further object is to provide mechanical wave filter sections having inherent therein impedance transformations as well as prescribed attenuation characteristics.

A further object is to provide composited mechemical vibratory systems in which parasitic transmission of energy is largely eliminated.

Other and further objects will appear during the course of the following description and from the appended claims.

The features of the invention will be more apparent when considered in connection with the accompanying drawings in which: i

Fig. 1 represents a three-dimensional bar of general shape, the nature and utility of-certain properties of which will be explained as related to the present invention;

Fig. 2 represents a differential element of a general bar;

Fig. 3 represents the equivalent electrical network of the general bar of Fig. 1 with an external impedance connected thereto at the point 0, of Fig. 1;

Fig. 4 is an alternate form of the network of Fig. 3;

Fig. 5 illustrates a simplification of the network of Fig. 3 efiected by centrally attaching the external impedance;

Figs. 6A, 6B and 6C, showthe side, top and end views respectively of a general rectangular bar of convenient shape for use in the devices of this invention;

Fig. 7 shows the details of a general cylindrical bar of convenient construction for use in the devices of this invention;

Figs. 8A to 10A, inclusive, and 83 to ICE, inclusive, represent variations of mechanical structures and their respective equivalent electrical networks, all of which are potentially equivalent;

Figs. 11A and 11B represent the mechanical and equivalent electrical structure, respectively, of a simple form of filter of this invention;

Figs. 12A to 12C, inclusive, illustrate the possibility of incorporating impedance transformation in a simple filter structure;

Figs. 13A and 13B represent the mechanical and equivalent electrical structure respectively of a second simple form of filter of this invention;

Figs. 14A to 14C, inclusive, represent more complex mechanicalstructures for filters of this invention;

Fig. 15A shows in diagrammatic form a filter system of this invention coupling two electromechanical transducers;

Figs. 15B and 150 show equivalent electrical structures of the system of Fig. 15A;

Figs. 16A and 16B showthe mechanical and equivalent electrical structures respectively of a confluent band-pass filter of this invention;

Figs. 17A to- 17C, inclusive, show the mechanical and equivalent electrical structures respectively of a confluent band-pass filter of this invention incorporating an impedance transformation;

Figs. 18A to 18C illustrate an alternate mechanical structure potentially equivalent to the combination shown in Figs. 17A to 17C, inclusive;

Figs. 19A, 19B and 19C to Figs. 21A, 21B and 21C, inclusive, Figs. 22A and 223 to Figs. 24A and 24B, inclusive, Figs. 26A and 2613, Figs. 27A, 27B and 27C, Figs. 28A and 28B, Figs. 29A and 2913, Figs. 32A, 32B and 32C and Figs. 33A, 33B, 33C and 3313 show further mechanical structures illustrative of the principles of this invention and indicate possible equivalent electrical networks for the respective structures illustrated;

Figs. 25A, 25B and 25C, Figs. 30A and 30B and Figs. 31A and 31B are employed to show the equivalence for particular purposes ofthe networks illustrated in each group, respectively;

Figs. 34A and 34B illustrate a method of supporting a mechanical filter structure of this invention;

Figs. 35 and 36 are employed to explain the principles involved in properly designing the frame to support a mechanical filter structure of this invention;

Fig. 37 is illustrative of the unit type of construction which may be employed in connection with filters of this invention; and

Figs. 38A, 3813, 39A, 39B and 40, inclusive, are illustrative of the application of the principles of this invention to a particular design of filter in which no vibratory energy is permitted to reach the supporting frame.

Figs. 41A, 41B, 41C, 41D, 41E and 41F show the equivalent electrical networks of another embodiment of the invention, the mechanical structure corresponding to these networks being shown in Figs. 42A and 42B, Figs. 42C, 42D and 42E illustrating elements used in the structure of Figs. 42A and 42B in greater detail.

The general bar theory underlying the filter sections of this invention may be explained as follows: Consider a solid body, as indicated in Fig. 1, of density p (at, y, z) in oscillatory motion such that the instantaneous velocity at x=Z1 is 2'1 parallel to, and positivein the sense of, the positive yaxis, While the instantaneous velocity at .l.'=l2 is i2 parallel to, and positive in the sense of, the negative y-axis. It is obvious that the motion at any point on the x-axis will be parallel to the y-axis and that the motion of the whole body at any instant will be one of rotation in the XY plane about some point on the m-axis as the instantaneous center of rotation. From the known velocities i1 and i2, the center of rotation X may be calculated. The equation of the line drawn from the tip of 'i1 to that of i2, shown as a dotted line in Fig. 1, is

concerning this line we know that when i=0, x=r (2) $=l1, i=+i1 (3) ;c=+l2, 12=i2 4) and z= zl+zg from which we find x.= 'j lt (6) Next the distance r from lie to a typical mass element dM at .13, y is found by the familiar relation and the velocity iiof this mass element is determined ini terms of i1, i2, Z1, Z2, m; :11. The final step is to compute the kinetic energy and inlowing manner:

Thekinetic energy is given by:

Substituting Equations 6 and 10 in Equation 9,

To obtain the equivalent network for a differential element, as illustrated in Fig. 2, we write the kinetic energy expression below:

dT= dL .i{+ dL12(i1i2) ulna; 2 2 2 Collecting terms dT=%(dL1+mm-dLailmwLfi ime (12) By direct comparison of Equation 11 and Equation 12,

If we add a mechanical impedance Z to oppose the motion of the body at point :1:=0, 11:0, replace lflvi by dxdydz and integrate symbolically over the whole volume, the following equations are obtained, where the'triple integrations are to te rateover all the mass elements in the- 1-.

bejextended overthe whole space occupied by the body. i

L f f f 1 01-2 311-22 +2x +2y da:dydz

As to the external impedance Z attached to the point on the body :r=0, y=0, the mechanical volt-amperes absorbed by Z are By a direct comparison, in a manner similar to that employed in connection with Equation 12,

. h z,= z 2c) The simplest and the most useful electrical equivalent of the system is that shown in Fig. 3.

With these difierences, Fig. 3 may be shown to be equivalent to the 3-mesh network of Fig. 4. However, this and numerous other alternative equivalents do not add appreciably to the utility of the general bar in mechanical frequency-selective or distortion-corrective networks. Hence, it is expedient in the interest of simplicity to return to the original equivalent with the first set of formulae, viz., (19) to (21) and (25) to (27), inclusive.

ydition wouldbe to have the body of Fig. 1 rotatlefore these formulae are simplified for practical applications it should be noted that, J

where M fff y where M is obviously the total mass of the body.

Moreover,

As an aid in the physical interpretation of the above results the following re-expression of formulae 19 to 21 and 25 to 27, inclusive, shoul also be noted.

By formulae (19) to (21), inclusive, we may write 1 arme- -2 -z,, 22 and are the :r-coordinates, of points 1,2 and the midpoint between them, respectively, and we may Write where 11, and I2 and 11 2 are the 11101118111501 in ertia about points 1, 2 and the mid-point between them respectively. 1 1 1 L1, L2 and L12 maybe expressed in terms of any three of the four parameters I1, I2, he and M, by way of examples, Equations 40 to 42, inclusive,

above, or Equations 43 to 45, or 46 to 48' below. 1

the effective mass would be ing about-the mid-point between points -Z1and Z2. 1 Under this condition, the moment of inertia is I1 2, the lever arm is andtheefiective mass is Similarly L1+L12 is the inductance, again assuming Z1 'Z2 and Z12 are not present, measured acrossterminals I and 2 of Fig. 3. In this case i2 would be zero 'andthe equivalent mechanical conditions would be'to have the body of Fig, 1

rotating about the point at 12. Under this condition, the moment of inertia is I2, the lever arm -I and theefiective mass is By the same process, a similar measurement across terminals .3 and 4 of .Fig. 3 in the absence of Z1, Z2 and Z12 would indicate the value of Lz-i-Lm; ii would be zero, the body would be rotating about the point at --Z1, the moment of inertia would be I1, the lever arm would be Z and 1 111a similar manner when the mechanic-a1 imped'ance Z has been added to the system, the efiective components Z1, Z2 and Z12 resulting in the equivalent electrical network of Fig. 3 may be expressed as These latter formulae have a physical significance analogous to that of formulae 34 and 40 to 42, inclusive, above.

If the point to which Z is attached is taken half-way between the points to which i1 and i2 are referred l (that is t =-l the electrical equivalent reduces to that shown in Fig. 5 where;

. 1 1 Z (57) The importance of this simplification lies in the fact that Z now no longer appears in the series armsp ,Thus, thechoice of z will depend only upon the desired impedance Z12 in series with "the inductanceL12 in the shun-t branch.- Any desired impedances in series with the 'inductances L1 and L2, respectively in the series'branches,

may now be supplied externally at the points 5 --l1 and 12 of the body, and these added impedances will be independent of Z.

By properly shaping the body, L1, L12, and L2 may, at least theoretically, be made to haveany preassigned set of values consistent with Equations 34, 35 and 36. It has been found, however, that a bar of uniform density and of the shape shown in Figs. 6A- to -60, inclusive, is sufficiently general for most practical cases. 1 The body. of the bar 45 is symmetrical with respect to each of the three coordinate planes and is of length 1. The horns, that is, the pieces 46 and 41 added at the ends of the barare symmetrical with respect to-the X-Y and the XZ planes, respectively, but not with respect to the 'Y--Z plane unless the desired values of L1 and L: for the composite bar are equal.

For design purposes this bar may be considered to be a composite bar of which each of the component parts 45,46 and His a simple rectangular bar although in practice the composite bar, may obviously be made in' one piece. The formulae for L1, L12 and L2 are applicable to each component part and the results are additive. Thus, it will be the usual, because it is thesimplest, procedure to design the body of the bar 45 to give the desired value of L111 and to supply the deficiencies in the values of L1 and L2 by designing the horns" 46 and 41 so as to contribute nothing to L12.

- The following formulae, by way of illustration, 'give the contributions of a single. rectangular component part bounded by the planes "45 where a 0, b 0, c and h and k are perfectly arbitrary. (See Figs. 6A to 6G, inclusive.)

Thus for the body of the bar 45, for which 70:0, a=l, the preceding formulae give 75' in attaining the desired AL12=0. It is necessary only. to choose k,'a, b and 0 so as to satisfy the following equations.

Since there are only three of these equations to be satisfied by the four design parameters there is obviously a single infinity of simultaneous values. of these parameters. Moreover, further latitude is furnished by the fact that the deficiencies in the values of L1 and L2 of the composite bar may be supplied by the two pairs of horns 46 and 41 one at each end of the composite bar, in any proportions that might prove convenient. (When AL1 AL2 and or when AL1 AL2 and the appendages 46 and 41 lose the character of horns). If, for example, the deficiencies are 10 in L1 and 7 in L2, these could be supplied by a single pair of horns for which Or, two pairs couldbe used for one of which AL1 is anything less than 10 and AL2 is anything less than 7, and the deficiencies are supplied by the other pair. The two pairs of horns 46 and 41 might or might not overlap. If no convenient dimensions can be found for which they do not overlap, they can as a practical matter be overlapped in the direction of the z-axis without invalidating thetheory. Usually a. few judicious trials will suflice to arrive at satisfactory dimensions for component parts of the composite bar.

The foregoing procedure whereby the horns are designed'to contribute nothing to the L12 of the composite bar affords a practicable, and usually al'so a practical, method of effectively adding point masses," that is, masses which may be considered to be concentrated at a single point, at the ends of a simple bar.

In some applications it may be necessary, in order to obtain more suitable or convenient dimensions for the composite bar, to design the body of the bar 45 to furnish more or less than the desired value of L12 and to design the horns 46 and 41 so as to compensate for the excess or deficiency in'Lm as well as the deficiencies in L1 and L2. In such cases the formulae to be satisfied by k, a, b and c arethe following:

L ALj-AL,

v i a e e an An important extension of the idea underlying the procedure outlined in the above paragraph is to design the horns 46 and 41'so-that the composite bar shall have zero or negative values of L12 when the equivalence and formulae appertaining to Fig. are used, in which case the horns will obviously have to extend beyond the ends of the body of the bar if the type of construction shown in Figs. 6A to 60, inclusive, is used. Zero values of L12 will be necessary, of course, when the desired shunt-branch impedance has a root instead of a pole at zero reciprocal frequency (1. e. at infinite frequency). The possibility of obtaining negative values of L12 opens up a field of filter characteristics to mechanical filters which is accessible to ladder-type electrical wave filters only through the use of electrical transformers to furnish the negative mutual inductances required.

In practice, it often happens that a composite bar possessing cylindrical symmetry, as shown in Fig. 7, is desirable. In such a bar the horns become cylinders such as 43 and 44 concentric with the body of the bar 42. Consider a typical component bounded by two concentric cylinders 42 and 43 of Fig. 7 having radii b and 0, respectively, where b c, with the x-axis as their common axis and by the two. planes Corresponding to the formulae (58) to (61), inclusive, the following formulae now hold:

Thus for a uniform cylindrical bar, 70:0, 0:0, a=l, whence In the following paragraphs illustrative examples will be given of the practical application of the general bar theory to the attainment of mechanical structures which possess dynamical properties corresponding to each of a list of wellknown ladder-type electrical filter sections. The sections selected for this purpose are mainly those listed on pages 315 to 318 of the book Transmission Networks and Wave Filters by T. E. Shea published in 1929 by D. Van Nostrand Co. Inc., 250 Fourth Avenue, New York city, and also some of the types described in an article entitled Mutual Inductance in Wave Filters, with an Introduction on Filter Design by K. S. Johnson and T. E. Shea in the Bell System Technical Journal, Vol. IV, No. 1 of January 1925, containing effectively negative inductances. The designations employed in the first-mentioned book by Shea for the various filter sections being discussed will, in general, be used in the following description.

The examples to be given hereunder by no means exhaust the number of mechanical structures of the present invention which possess dynamical properties corresponding to each of the electrical sections listed either individually or in combinations as components of a composite structure. Neither do they by any means constitute a complete list of the electrical networks whose dynamical characteristics may be simulated by mechanical combinations embodying the principles of this invention.

An attempt to exhaust the practical applications of the present invention to the design of mechanical vibratory systems having useful properties would be extremely tedious and will not be made in this application. A large number of structures will, however, be described in an effort to adequately illustrate the extremely broad scope of the principles underlying the invention.

The structures here described have been chosen with a View to bringing out some of the more important contingencies which are encountered in the design of several novel types of mechanical structures suitable for use in composited mechanical filters. Limitations which may he possessed by particular structures here described are not indicative of inherent limitations in structures of the invention.

It should be borne in mind that in the case of mechanical filters of this invention, as in that of electrical filters, there is a Wide choice of structures capable of giving any particular set of transmission, phase and impedance characteristics, provided there is any. It is for this reason that it is expedient to distinguish between electrical analogs and electrical equivalents.

The distinction will be apparent from the following illustration. Under ideal conditions the electrical analog of the conventional mechanical filter of Fig. 8A is shown in Fig. 8B, that of Fig. 9A is shown in Fig. 9B, and that of Fig. 10A is shown in Fig. 103. But the electrical networks of Figs. 8B, 9B and 10B are potentially equivalent since by substitution of equivalent T and 1r con- .denser combinations (see page 91 of Sheas book,

supra) they may all be readily reduced to the form shown in Fig. 8B. Hence, Fig. 83 may be regarded as the electrical equivalent of the mechanical structures of Figs. 9A and 10A as well as the electrical analog of the structure of Fig. 8A. Bearing this in mind, it is clear that the general bar has no electrical analog; but it has an infinite number of electrical equivalents, of which two are shown in Figs. 3 and 4, respectively.

Any hypothetical electrical network possessing the same transmission characteristics as a mechanical structure employing bars cannot, therefore, be any more than. an electrical equivalent.

This being so, any particular electrical equivalent may be transformed into any other equivalent electrical network which latter may have a greater or smaller number of meshes and which may or may not contain ideal transformers. By such artifices, a particular mechanical system may be shown to have particular filter characteristics, either per se or in conjunction with other mechanical structures, and the mechanical system may be shown to possess in some instances internal impedance transformations.

In the case of electrical filters, the choice of structureis determined principally by the cost of the elements." While ordinarily the structures requiring the minimum number of elements are least expensive, situations are sometimes encountered in which the choice of a structure containing more elements results ina saving in cost. In the case of mechanical filters the cost of the parts is usually small and the choice of structure is, therefore, usually almost entirely dictated by considerations of mechanical construction, facility of adjustment and likelihood of accidental maladjustment. Also, some types of construction are capable of yielding higher effective efficiencies than others.

In the diagrammatic representations of mechanical structures in the accompanying drawings, stiifness or elastance members, that is, either reeds, leaf springs or taut wires, are represented by simple lines. Usually where taut wires are employed, two conveniently spaced parallel or converging wires may be used to advantage to facilitate the mounting of the bars thereon and to constrain the mechanical system from vibration in undesired directions. Composite bars of the type shown in Figs. 6A to 60, inclusive, or that shown in Fig. 7, are represented by simple rectangles and fixed parts, such as points of attachment to a fixed supporting frame, are represented by hatchings designated F. Auxiliary tensed elastance members, which may also carry bars, are provided in numerous instances and are placed in a common plane with the elastances and bars of the main system but with their longitudinal axes in approximate quadrature relation with those of the main system. Leaf springs of approximately semicircular form are usually employed to couple electromechanical transducers to appropriate points of the mechanical system for imparting or withdrawing energy therefrom. In a few instances such springs are also employed to provide additional elastances as parts of the mechanical system per se. In general, leaf springs are mounted in the plane of the vibratory motion rather than in a plane at right angles thereto as in the case of tensed wire elastances.

'To facilitate visualization of the mechanical combinations employed in the systems of this invention, many of the representations will be shown in quasi-perspective diagrammatic form. A bar designed to have zero mutualmassis distinguished from others by a small circle at the center of the rectangle. Small arrows designated A and B are employed, in the absence of other indicative structure, to jShOW'thB appropriate points at which energy should be introduced into or drawn from the mechanical vibratory system in order that its characteristics with respect to mechanical energy transmission should be comparable to the electrical characteristics of the equivalent electrical network indicated.

Although the familiar type of mechanical filter construction illustrated in Figs. 8A to 10A, inclusive, the sections illustrated having the potentially equivalent electrical structure designated 1111 by Shea, supra, is practicable in some instances when this type of section may be used exclusively, it is difficult, and at times impossible, to join other types of sections with it in a composited structure. Furthermore, the masses, such as M1 and M2, of such structures cannot physically be made either'actual or equivalent point masses and each mass element provides only. one reactive component to the formation 35 @of the mechanicalivibratory structure. I Bearing these facts in mind, it is apparent that the structure of Fig. 11A, embodying the principles of the present invention, lends itself more readily and ,far more advantageously to use in composited mechanical filter structures. The structure of Fig. 11A has the added advantage of being more rugged mechanically and less subject to the hazard of being thrown out of adjustment.

In Fig. 11A, tensed wires 52 and 53 are mechanically equivalent to the series condensers 52' and 53' of the electrical network of Fig. 113. Bar of Fig. 11A, having zero mutual mass, provides only point masses, one at each point of support, equivalent to inductances 8| and 82 of Fig. 113. The auxiliary tensed Wire 5| of Fig. 11A provides the only mutual or shunt impedance of the system and is the equivalent of condenser 5| of Fig. 113. Auxiliary wire 5I, as mentioned above, is placed in the same plane as wires 52 and 53 but with its longitudinal axis at a suflicient angle (usually 90 degrees) to avoid interference with other parts of the system.

It should be noted that the type of construction shown in Fig. 11A may be designed to provide impedance transformation. This becomes evident if we start with the network Fig. 12A, transform it into the network Fig. 12B and, making use of the equivalence shown in Fig. 67, page 136 of the above-mentioned book by Shea, further transform it into the network of Fig. 12C, which is identical in form to thato-f Fig 113.

In the practical design of a similar electrical section with impedance transformation, the transformation ratio is limited by the fact that the impedance transformation ratio s must be positive in order that the shunt stiffness. or capacity shall be positive and further by the fact that negative values of the stiffness in the righthand series branch must be numerically less than the positive stiffness in the left-hand series branch of theadjacent section on the right-hand side.

This limitation obviously holds in the case of the mechanical section of Fig. 8A and also in that of Fig. 11A, if the cross-wire in the latter a is midway between the. ends of the bar. However, this median location of the impedance to motion of an internal point of the bar was introduced, as mentioned above, merely in order to simplify the formulae. By going back to the more general conditions under which the original formulae I9 to 2I, inclusive, and 25 to 21, inclusive, prevail, the limitation imposed by the sign of the. stiffness in the series branch is, to a large extent nullified. It is then possible to effectively obtain negative stifinesses in one of the series branches.

A mechanical equivalent of the section designated IIIz by Shea, supra, is provided by the structure of Fig. 13A, as is obvious from its electrically equivalent structure shown in Fig. 13B, the bar 59 of Fig. 13A providing the mechanical equivalent of the T-network of inductances I02, I03 and I06 of Fig. 133 and the elastance members 58 and 60' providing the mechanical equivalents of the condensers IUI and I04, respectively.

Asin the case of the structure of Fig. 11A the structure of Fig. 13A may be designed so as to provide impedance transformation. This may readily be seen by interchanging capacity for inductance and vice versa in the network shown in Figs. 12A to 12C, inclusive.

While it may superficially appear that the transformation ratio is limited by the impossibility of obtaining negative inductance in the series branches, actually there is. no limitation since the ratio may in this instance be negative. Thus, it is not necessary withthis type of section to relinquish the simplification obtained in Fig. 5.

While the constant-K confluent type of section, designated IVk by Shea, supra, is equivalent to the combination of a III1 type and a IIIz type section designed to the same image-impedance and connected in tandem, it is not practical to join the structures shown in Figs. 11A and, 13A since this would require an ideal hinge, that is a hinge having no friction and no clearance. However, if these structures are joined through a mutual stiffness as may be done in a number of ways, two examples being shown in Figs. 14A and 143, respectively, the combination is easily shown to be potentially equivalent to, two 11112 and one III1 type, or to one constant-K confluent and one 1112 type section, with or without impedance transformation. The structure of Fig. 140 illustrates a third method which gives th same result.

The above paragraph illustrates a difficulty which not infrequently arises, when it is desired to join two mechanical structures which are satisfactory when used separately. The result may in some instances be somewhat unsatisfactory from a practical standpoint but can often be remedied by slight modifications which may introduce additional sections into the electrical equivalent. These additional sections may fortuitously fill a desired purpose which would otherwise have to be met by coupling still another mechanical structure to the system, as for example, a second structure like Fig. 11A, if the first and the IlIi type section were joined by an ideal hinge. On the other hand, of course, the additional sections may, for reasons such as delay or distortion, pro-- duce in some cases an undesirable composited structure.

Assuming that the additional III2 type section isv tolerable or desirable, there are simpler mechanical structures than those of Figs. 14A to 140,

inclusive, which will give the same result. An

' ployed as an input device to convert electrical into mechanical energy and apply it to the mechanical vibratory system and the other serving as an output device to convert the mechanical energy received by it from the mechanical vibratory system into electrical energy. The moving coils I40 of these transducers are carried on diaphragms I II clamped at the edges in fixed supports F. Capacities I41 and I54 of Fig. 15B are the electrical equivalents of the diaphragms I4I of the input and output transducers, respec-- tively, and inductances I46 and I56 are similarly the electrical equivalents of the masses of the respective moving coils and associated parts of the input and output transducers. Shunt capacities I46 and IE3 of Fig. 15B are the electrical equivalents of elastance members I38 and I 43, respectively, of Fig. 15A and the T-network of inductances I50, I5I and I 52' of Fig. 153 is the electrical equivalent of the bar I42 of Fig. 15A. The equivalent network of Fig. 15C may now be readily derived from Fig. 15B by reversing the inductance L-network formed by coils I50 and I5I of Fig. 15B and the capacity L-network formed by condenser I53 and effectively a portion of condenser I54 of Fig. 15B. That is, condenser I54 may be replaced by two condensers in series, one of which will be associated with the L-network to be reversed and the other of which will provide the desired capacity I62 of Fig. 150. The principles underlying the reversal of L-network, as above suggested, are explained in Sheas book, supra, at pages 135 and 136. An impedance transformation is introduced in each instance and the two may compensate wholly or partly for each other or they may be additive so that the impedance at terminals 3, 4 of the network of Fig. 150 may be equal to or larger or smaller than that at terminals I, 2 of the network.

As a matter of design practice, the electrical network of the form of Fig. 15C, which would provide the desired attenuation characteristics and terminal impedances, may, if convenient, first be V determined. From it an equivalent network of theform of Fig. 153 may then be derived and the mechanical system of Fig. 15A may then be designed to be the mechanical equivalent of the network so derived.

A mechanical structure which provides a con stant-K confluent section more directly than the combination of a III1 type and a IIIz type section worked in tandem, is that shown diagrammatically in Fig. 16A. An equivalent electrical network of the structure of Fig. 16A is shown in Fig, 163, condensers I10 and I16 of Fig. 16B being the equivalent of elastance members I63 of Fig. 16A, inductances HI and I14 of Fig. 16B being the electrical equivalent of bar I68 of Fig. 16A and inductance I12 and capacity I13 of Fig. 163 being the electrical equivalents of bars I66 and elastance members I61 of Fig. 16A, respectively.

It should be noted that all three bars of Fig. 16A are shown diagrammatically (by the small central circle) as having no mutual masses. Because of this, bar I68 contributes no shunt inductance and bars I66 isolate elastance members I64 and the points on the supporting frame to which they are connected from the vibratory energy of the system and provide effectively point masses.

Elastance members I61 must, of course, as explained above, be connected at the mid-point of bar I68 so that the impedance provided by the combination of elastance members I61 and bars I66 will appear solelyin the shunt arm of the equivalent electrical networks.

Energy is introduced into the mechanical vibratory system at one end of bar I68 and taken from it at the other end of this same bar. Elastance members I63 are, of course, active compon-ents of the vibratory system and hence vibratory energy may through them reach their respective points of support on the supporting frame of the system. Measures to be taken to avoid possible deleterious effects, such as parasitic transmission of energy through the frame or the dissipation of energy in the frame will be discussed at length in connection with other structures illustrative of the principles of this invention which will be discussed hereunder. It should be noted that no impedance transformation is incorporated into the structure of Fig. 16A.

Two mechanical structures embodying the principles of this invention which provide a constant-K confluent section having impedance transformation are diagrammatically shown in Figs. 17A andlBA.

In the structure of Fig. 17A, bar I78 is designed to have no mutual mass so that elastance member I I1 is isolated from the vibratoryenergy of the system. Bar I18 and elastance mem ber I88 of Fig. 17A are represented in the equivalent electrical network of Fig. 173 by inductance I86 and capacity I 84, respectively. Elastance members I8I and I83 of Fig. 17A are represented by capacities I81 and I92 of Fig. 173, respectively, and bar I82 of Fig. 17A is represented by the T-network of inductances I88, I and I9I of Fig. 17B. The network of Fig. 173 may then be shown to be equivalent to that of Fig. by inverting the L-network of inductances I38 and I90 in accordance with principles, above explained, whence it becomes apparent that the structure of Fig.1'7A may be designed to provide a mechanical structure having as an equivalent electrical network a constant-K confluent section and an impedance transformation.

Similarly, the structure of Fig. 18A comprising elastance members 286, 291 and 2H and bars 298 and 2IIl may obviously be designed to have an equivalent electrical network of the type shown in Fig. 18B which may in turn obviously be equivalent to "a network of the form shown in Fig. providing for impedance transformation at each end of the network.

The elimination of one of the shunt stiffness elements in Fig. 15A, that is, coupling one of the transducers directly to the near end of bar I 42, also leads to a constant-K confluent section. The resulting structure then becomes essentially a tandem combination of a K12 type section of Fig. 8A and a III1 type section of Fig. 13A. However, it is not advisable to use a structure which requires the rigid attachment of the armature or moving coil of an electromechanical transducer, hereinafter referred to as a receiver, to a bar since the oscillatory motion of the bar may be' impeded by the armature or moving coil, and the armature or moving coil will experience a twisting motion about the instantaneous axis of rotation of the bar.

Additional structures which provide the equivalent of two I112 type and one III1 type sections in tandem, or one constant-K confluent and one IIIz type section in tandem are derivable from the structures of Figs. 16A to 18A, inclusive, bv the obvious expedient of interposing a mutual stiffness between either the input or the output point and the corresponding receiver, advantage being taken of the inherent mass and stiffness of the receiver armature 'or coil and diaphragm for the final series branch. Other expedients, corresponding one to each of the structures of Figs.'l6A to 18A, are shown in the structures of Figs. 19A to 23A, inclusive. The equivalent electrical networks are obviously those of Figs; 19B, 19C, 20B, 20C, 21B, 21C, 223 and 23B, respectively.

The principles and processes involved in demonstrating the relations between the several forms of equivalent electrical networks for the mechanical structures disclosed in these figures. and the further illustrative figures which will be described hereunder, should now be obvious from the several detailed discussions given above, such as those relating to Figs. 15A, 15B, 15C, 16A, 16B, etc. As it -is felt that no difficulty will be encountered by those conversant with the art in perceiving the equivalences in the remaining figures, the description thereof has been simplified by the omission of the detailed identification of each mechanical element with each electrical element and the detailed description of transformations, above described, and well known to be applicable in deriving one electrical network from another in the several manners indicated.

Structures which provide the equivalent of one IIIz type and two III1 type sections in tandem, or one constant-K confluent section and one III1 type section in tandem are derivable from the structures of Figs. 14A and 18A as illustrated by the structures of Figs. 22A and 23A. It should be apparent from the above discussion that a number of other unique mechanical structures of this invention may also be readily derived which will give the same results.

The m-derived sections, designated 1V1 and W2 by Shea, supra, are obviously provided by structures similar to those of Figs. 11A and 13A. A particular example is shown in Fig. 24A, one equivalent electrical network being shown in Fig. 243. The structure of Fig. 24A obviously may possess internal impedance transformation. As in the case of electrical ladder sections which make use of transformers, these m-derived structures are not restricted to positive values of m within the range from 0 to 1 only, but may have any positive value of m within much wider practical limits. This comes about from the fact that the mutual mass of the bar may be negative when the mutual stiffness is positive, or vice versa, depending upon the choice of conventional positive directions inthe meshes of the electrical equivalent.

The practical limits on the value of m are dependent only upon the practical limits on the coefficient of coupling of the bar in the case of m-derived IIIz type sections, and on the cutofi's as well in the case of m-derived I111 type sections. In order to show the form of these re lationships it is necessary to observe, first that the lattice network of Fig. 25A is equivalent to the T-networks of Figs. 25B and 25C if phase shifts of 180 degrees, independent of frequency, are disregarded. In the case of m-derived IIIz type sections,

It is obvious then that if m 1 (the peak of attenuation occurs between the upper cut-off and infinite frequency) the T-network of Fig. 25B should be used in design work. If m f2/f1 (the peak of attenuation occurs between the lower cut-oil and zero frequency) the T-network of Fig. 250 should be used. If 1 m fz/f1 (the peak of attenuation occurs at no real frequency, zero'and infinity included) and the T-nctwork of Fig. 25B should be used.

Denoting the coeflicient of coupling in Fig. 253 by RB,

1k 1+kB while if he be the 'coeflicient of coupling in Fig. 25C

whence -H c' WA-kc The limitations upon the value of m imposed by practical considerations concerning the rigidity of the bar may be illustrated by the following example. Suppose that the coupling coefficient should be confined to the limits 0.6 and +0.8. In the case of Fig. 25B, 121. will be confined to the limits 1/3 and 2, while in the case of Fig. 250, m will 'be confined to the limits 1/2 and 3. The minwhence mum value of foo/f2, when the peak is above the I upper cut-off, is then determined by f f1 Thus if f2 3f1 it is not practical to place the peak anywhere below the lower cut-off, while if foo /f1=0.94 is to be practical I: must be less than In the preceding paragraph it was stated parenthetically that when the value of 'm for an m-derived IIIz type section was greater than fz/fi, the peak of attenuation occurs between the lower cut-off and zero frequency. This follows from the essential equivalence of the m-derived I111 and IIIz type sections. Except for a constant phaseshift of 180 degrees, independent of frequency, the values of m in equivalent m-derived III1 and H12 type sections are related by the equation where m1 and me relate to 1111 and H12 type sec tions, respectively. Thus, when mz f2/f1,m1 1';' when mz 1, m1 f2/f1; and when l mz f2/fr, 1 m1 f2/f1. The foregoing paragraphs, therefore, really dispose of the m-derived III: type section with m 1 as a special case of the m-derived I112 type section with m f2/f1. If the same analysis is carried out on the m-derived IIIi type section, the physical limitations upon the loca tion of thepeaks of attenuation must of necessity turn out the same.

It may also be noted that for the m-derived III1 or III: type section where 1 m- .f2/f1, the peak of attenuation occurs, mathematically speaking, on the imaginary axis of the frequency plane. At real frequencies the attenuation remains finite even for zero and infinite frequencies.

With respect to the five-element sections, designated V1 and V2 by Shea, supra, the V1 section is equivalent to a III1 type section and an m-derived III2 type section. Hence, if an additional 1H2 type section is desired, or if it may be tolerated, a structure of the type of Fig. 13A may be joined to one of the type of Fig. 24A through a mutual stiffness. If the additional section introduced' by this type of coupling is not desirable the structure illustrated in Fig. 26A may be used.

The structure of Fig. 27A, which is an obvious modification of that of Fig. A, provides a V2 section and an additional IIIz type section. If the additional section is not desirable, the structureof Fig. 28A may be used.

The six-element dissymm'etrical and six-elementsymmetrical sections designated VI1 by Shea, supra, are equivalent to a tandem combination, of a 1V1 and a IV2 section. However, two mechanical structures of the type shown in Fig. 24A cannot be coupled practicably except through a mutual stiffness which adds a K12 type section to the composited filter characteristic. If the additional section is desired this is an excellent way of, securing it. If undesirable, other mechanical structures must be sought.

A type of mechanical structure which provides a six-element (in'general, dissymmetrical) 'section directly is that shown in Fig. 29A. It bears a structural similarity to the IVk structure of Fig. 16A. Design formulae directly applicable to the electrical equivalent shown in Fig. 293 will be found in Fig. 148, page 269 together with formulae (108) page 267, (112) page 269 and (125) and (126) page 275 of Sheas book, mentioned above.

The above descriptions have been confined to the mid-series type sections, that is, sections which have the same mid-series image-impedance as a constant-K or Wk type section. Since the principles involved are quite similar only a brief presentation of the mid-shunt type sections will be given.

, Taking the electrical network of Fig. 153 and carrying it through the transformations indicated in Figs. 30A and 30B, it is apparent that the mechanical structure of Fig. 15A is potentially equivalent to a half. 1V1; section with the midseries end towards one receiver, a full mid-shunt 1113. section, and another half IVk section with the mid-series end towards the other receiver. Similarly, taking the electrical network of Fig. 23B and carrying it through the transformations indicated in Figs. 31A and 313 it is apparent that the mechanical structure of Fig. 23A is potentially equivalent to a half We section with the midseries end towards one receiver, a full mid-shunt I114 section, and another half IVk section with the mid-series end towards the other receiver.

Considering next the mechanical structure of Fig. 32A, it is obvious from the equivalent electrical networks of Figs. 32B and 32C, potentially equivalent to two half IV}; sections with their 

